simply connected covering of a path connected space Let $p:\overline{X}\rightarrow X$ be a simply connected covering of a path connected space $X$ and $A\subset X$ be a path connected set.
Show that the inclusion induced homomorphism $i_{\sharp} : \pi_1(A)\rightarrow \pi_1(X)$ is surjective iff $p^{-1}(A)$ is path connected.
Assume that $i_{\sharp} $ is surjective. Let $\bar{x},\bar{y}\in p^{-1}(A)$ then we have $p(\bar{x}),p(\bar{y})\in A$. 
As $A$ is path connected we have a path $\omega : I\rightarrow A$ such that $\omega(0)=p(\bar{x})$ and $\omega(1)=p(\bar{y})$.
As $p$ is a covering projection we can lift this to a path in $\overline{X}$
There exists $\eta : I\rightarrow \overline{X}$ such that $p\circ \eta =\omega$. So, we have $p(\eta(I))=\omega(I)\subset A\Rightarrow \eta(I)\subset p^{-1}(A)$.
So, we see that the lift is actually in $p^{-1}(A)$ we also know that $\eta(0)=\bar{x}$ but it is not clear why should $\eta(1)=\bar{y}$.
I have used just that $A$ is path connected and nothing else...
 A: Using the surjectivity of $\pi_1(A,x_0)\to\pi_1(X,x_0)$, one can show that every path in $X$ between two points in $A$ is homotopic to some path in $A$. This this end, let $x$ and $y$ be points in $A$. There is a path $\alpha$ from $x$ to $y$ in $A$. Let $\beta$ be any path from $x$ to $y$ in $X$. Then $\beta \simeq \lambda\cdot\alpha$ for some loop $\lambda$ at $x$, and this $\lambda$ is homotopic to some $\kappa$ in $A$. Therefore, $\beta$ is homotopic to $\kappa\cdot\alpha$ in $A$.
Now let $\tilde x$ and $\tilde y$ be two points in $p^{-1}(A)$ with $x=p(\tilde x)$ and $y=p(\tilde y)$. Since $\tilde X$ is path connected, there is a path $\sigma$ from $\tilde x$ to $\tilde y$. Then $p\sigma$ is a path from $x$ to $y$, and thus homotopic to some path $\omega$ in $A$. Since $p$ is a covering map, their respective lifts $\sigma$ and $\eta$ at $\tilde x$ must be homotopic as well, and that means that $\eta$, a path in $p^{-1}(A)$, ends at the same point as $\sigma$, namely $\tilde y$.
The simply-connectedness of $\tilde X$ is only relevant for the proof of the other direction.
